1.1Estimating flow duration curves in ungagged catchments
The flow duration curve (FDC) is defined as the relationship between a certain flow rate and the frequency greater than or corresponding to that flow rate during certain periods of time. It is essentially a cumulative distribution function (Searcy, 1959), which comprehensively describes the entire characteristics of runoff in a basin from low flow to flood, and can better reflect the precipitation and runoff conditions of the basin (Cheng et al., 2012). However, many water resource projects are often located in areas without measured runoff data which leads to difficulty to directly obtain the FDC (Veber Costa, 2020). Regional analysis methods can be used to obtain the regional FDC from the areas with measured runoff data, and convert it to areas without measured runoff data to meet the design needs of water resource projects in that area, which is of great help for water resource planning, design, and runoff prediction in areas without measured data (Manuel Almeida, 2021; Veber Costa, 2020; Mancini, 2016; Li et al., 2010; Croker K M, 2003).
There are usually two existing methods to establish FDC in unmeasured areas: process and statistical based method (Blöschl, 2013) Based on process-based method, the probability distribution of daily flow is simulated by establishing a hydrological model of the watershed, considering the characteristics of the watershed and the physical mechanisms of the hydrological process (Cheng et al., 2012; Yokoo and Sivapalan, 2011; Ceola et al., 2010; Botter et al., 2007; Doulatyari et al., 2015) Statistical based method, on the other hand, are modeling methods based on statistical models and data analysis. It infers future hydrological variables by analyzing the statistical characteristics and patterns of historical observed data, which does not require in-depth understanding of the physical mechanisms of hydrological processes and typically require a large amount of data for training and optimization (Burgan and Aksoy, 2022a; Müller et al., 2014; Atieh et al., 2017a). The advantage of process-based models is that they can analytically derive the probability density function of flow and independently simulate the impact of climate or geomorphic changes on FDC, with reliability under non-steady conditions (Ghotbi et al., 2020; Ghotbi et al., 2020) . Its disadvantage is that the assumption of spatial homogeneity in the watershed makes its applicability relatively low (Leong and Yokoo, 2021) . However, parameter estimation for process-based models is less demanding, and can be determined using information such as rainfall, climate, and geomorphic characteristics of the watershed at any location with data (Schaefli et al., 2013; Karst et al., 2019). Based on existing analysis and research on process-driven method, the physical characteristics of the basin (such as average temperature, potential evapotranspiration, elevation, etc.) distribute precipitation to various parts of the river: groundwater recharge and base flow, surface runoff, and rainstorm flow (Rice and Emanuel, 2017; Ye et al., 2012) . Therefore, the precipitation accumulated and the features of the basin will mainly influence the shape of FDC (Luan et al., 2021).
However, due to the uncertainty in runoff and climate mechanisms, this method has limited application in areas without data (Reichl and Hack, 2017) . In addition, statistical methods often have better estimation performance on FDC than process-based methods (Engeland and Hisdal, 2009; Over et al., 2018) . It mainly include (1) Using regression methods to independently estimate quantiles through basin characteristics (Farmer and Vogel, 2016) (2) Estimating statistical moments and fitting the FDCs using appropriate distribution functions, finding the relationship between the statistical parameters of the function and the basin features (Almeida et al., 2021; Burgan and Aksoy, 2022b; Shin and Park, 2023) ; (3) Using streamflow index-based method (Atieh et al., 2017b). (4) Using geostatistical method ,etc (Goodarzi and Vazirian, 2023) . By comparing these two methods, it has been found that the statistical approach is more sensitive to spatially sparse data, while the process-based approach is more sensitive to observations that are temporally limited (Müller and Thompson, 2016). Although statistical methods often provide better FDC predictions than process-based methods, it is obvious that they typically need a significant amount of post-processing to explain the physical untrustworthiness in the results. There have been numerous studies attempting to combine process-based models and data-driven models in hopes of fully leveraging their respective advantages to improve prediction accuracy. For example, the relationship between quantiles and watershed features was studied to ensure the monotonicity of quantile estimation and explore the relationship between quantiles and their related basin features (Requena et al., 2018; Poncelet et al., 2017) .